Arrays of transducers are commonly used in such diverse fields as radio astronomy, seismic exploration, sonar, radar, communications and ultrasound imaging. The primary function of an array is to transmit and/or receive electromagnetic or acoustic energy over a specified region of space. Individual array elements are arranged along a line in a linear array, across a surface in a two-dimensional array or around a volume in a three-dimensional array.
The direction of energy propagation is controlled by introducing phase shifts and weighting to the signals delivered to and received from the individual array elements, so that signals transmitted to or received from the desired region in space constructively interfere while signals outside of this region destructively interfere. How well an array achieves this constructive and destructive interference is described by the radiation pattern of the array.
The radiation pattern is plot of the amplitude of the signal transmitted or received by the array as a function of position in space. In many situations the same array is used to both transmit energy and receive energy, and in these cases it is more useful to describe a transmit-receive radiation pattern, which is defined by the product of the transmit and receive radiation patterns. The transmit-receive radiation pattern gives a measure of the sensitivity and resolution with which the array will be able to detect objects in its field. Usually the transmit-receive radiation pattern is plotted in polar coordinates at a given distance in front of the array.
An example of a typical transmit-receive radiation pattern is shown in FIG. 1. The radiation pattern consists of a prominent main lobe and a number of secondary lobes. The main lobe corresponds to the desired region in space over which energy will be transmitted and from which energy will be received. The width of the main lobe is inversely proportional to the width of the array and determines the resolution of the array. Secondary lobes are caused by imperfect destructive interference outside of the desired region in space and result in the transmission and reception of unwanted energy. Thus, given a fixed number of array elements, a major problem which must be resolved when designing an array is how to minimize the width of the main lobe while keeping the secondary lobes as small as possible.
To optimize the array performance it is often useful to vary the weighting of the individual array elements. This is referred to as "apodization". The aperture of an array is given by a function which represents the element weighting as a function of the element position, as shown in FIG. 2 which illustrates as an example the receive and transmit aperture functions for a 6 element array with one-half wavelength (.lambda./2) element spacing. Separate aperture functions are defined for the array when it is transmitting energy and when the array is receiving energy, and each element in the transmit and receive aperture functions is represented by a delta function with an amplitude corresponding to the element weighting. In this example, identical element weighting has been used for each element.
The effective aperture E(x) of an array that both transmits and receives energy, a so-called "pulse echo" or "transmit-receive" array, is defined by the convolution of the transmit A(x) and receive B(x) aperture functions: EQU E(x)=A(x)*B(x)
where the symbol * denotes the mathematical operation of convolution. As can be seen in FIG. 2, with transmit and receive apertures having uniform weighting the effective aperture is triangular and has a width equal to twice the width of the individual transmit or receive aperture function. The transmit-receive radiation pattern of the focused array is given by the Fourier transform of the effective aperture. Thus, the beam pattern of the focused array is completely defined by the effective aperture of the array and, conversely, the effective aperture exhaustively defines the parameters for the array.
There are two main classes of arrays: periodic arrays and aperiodic arrays. In a periodic array, the elements are equally spaced. This is the most common form of array and the easiest to design, and there are a number of methods available for obtaining the minimum main lobe width for a given maximum secondary lobe pattern. However, the periodic arrangement of array elements creates additional unwanted main lobes called grating lobes.
The angular displacement of the grating lobes is determined by the distance separating adjacent array elements. To eliminate grating lobes in a periodic array it is necessary to space the elements no further than approximately one half wavelength (.lambda./2) apart, but an array that satisfies the .lambda./2 condition, known as a "dense" array, requires a large number of array elements. This can lead to unacceptable array complexity and cost, particularly for two- and three-dimensional arrays. Close spacing between elements can also lead to undesirable mutual coupling between adjacent elements, in which the signal from one element is distorted by the proximity of adjacent elements.
Arrays which have fewer elements than required to satisfy the .lambda./2 condition are often referred to as "sparse" arrays. Eliminating the grating lobes in a sparse array requires elimination of the periodicity of the array. This can be accomplished by varying the separation between different pairs of array elements, however large secondary lobes can still be present.
Designing an aperiodic array to minimize secondary lobes is difficult. A number of algorithms for selecting the element spacing in an aperiodic array have been proposed. However, it has been shown that sparse arrays designed by these algorithmic procedures were no better and often worse in terms of peak secondary lobe levels than sparse arrays in which the location of the array elements were selected at random.
More recently, computer optimization methods have been used to design the array geometry and element weighting to minimize a cost function which defines the desired relationship between the number of elements, the main lobe width and the peak side lobe levels. It has also been suggested that optimization methods which are used by adaptive arrays to remove interference or compensate for blocked elements could be applied to the design of maximally sparse arrays.
An alternative approach to minimize the number of array elements while reducing the secondary lobes has been proposed by von Ramm et al in "Grey Scale Imaging Photo-opt Inst. Engineers, Medicine IV, vol. 70, pp. 266-270, 1975. Von Ramm et al showed that the peak secondary lobes could be reduced by using different transmit and receive array geometries. They demonstrated that for a 16 element linear array a 7 dB improvement in the peak side lobe levels could be obtained when the inter-element spacing of the receive array was reduced to one-half that of the transmit array.
In "High Speed Ultrasonic Volumetric Imaging System--Part I: Transducer Design and Beam Steering", IEEE Trans. Ultrason., Ferroelect. Freq. Contr., vol. 38, pp. 100-108, 1991, Smith et al applies this idea to the design of a two-dimensional array in which the transmitter elements and the receiver elements are arranged in two cross patterns with the transmit cross oriented at 45.degree. relative to the receive cross. Smith et al taught that by arranging the elements in this manner, the secondary lobes in the receive radiation pattern would be located at nulls in the transmit radiation pattern and similarly, the secondary lobes in the transmit radiation pattern would be locate at nulls in the receive pattern. Using an array with 32 transmit elements and 32 receive elements, they obtained a secondary lobe level that was 15 to 20 dB below the main lobe.